Bulk-boundary correspondence in topological electrodynamics: From 1D to higher dimensions
ABSTRACT
A cornerstone of topological electrodynamics is the bulk-boundary (or bulk-edge) correspondence principle (BBCP): the number of topologically protected interface modes between two periodic heterostructures depends on discrete invariants (Zak phases in 1D and Chern numbers in higher dimensions) of the respective Bloch bands. In physics literature, analyses are typically performed only for Bloch modes in the bulk. Thus, conclusions about the boundary behavior of fields are reached after ignoring this behavior in the first place; this needs to be explained. Moreover, a puzzling feature of the BBCP is that the properties of evanescent modes in a band gap somehow depend on the properties of propagating modes at completely different frequencies.
The talk includes a full mathematical analysis of the BBCP and results for problems in 1D and “1.5D” (two-component fields but material parameters depending on one coordinate only); monotonicity of Bloch boundary impedance in the bandgaps plays a special role. The 2D case is qualitatively more complex, as scalar impedance turns into a Neumann-to-Dirichlet operator, which is also shown to be monotone in the gaps. Further mathematical considerations and numerical evidence are presented. Notably, for materials with frequency-dependent parameters, the BBCP turns out to be closely connected to the positivity of electromagnetic energy density.